This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 16b Thomas Scanlon Autumn 2010 Thomas Scanlon Math 16b Chain rule recalled (f ยท g ) (x ) = f (x )g (x ) + f (x )g (x ) Thomas Scanlon Math 16b Inverting the Chain Rule: Integration by Substitution (f ยท g ) (x ) = f (x )g (x ) + f (x )g (x ) f (x )g (x ) = (f ยท g ) (x ) โ f (x )g (x ) f (x )g (x )dx =
(f ยท g ) (x )dx โ g (x )f (x )dx g (x )f (x )dx f (x )g (x )dx = f (x )g (x ) โ Thomas Scanlon Math 16b Formalism of Integration by Parts Often, one nds two functions u and v so that the integrand may be written as udv where dv = v (x )dx . If we succeed in so doing, then from the equality u (x )v (x ) = u (x )v (x ) โ v (x )u (x )
we see that if f (x ) = u (x )v (x ), then f (x )dx = u (x )v (x )dx = udv = uv โ vdu If vdu is easier to evaluate than is succeeds.
Thomas Scanlon Math 16b f (x )dx , then the method An integral revisited Take u = x and v = โ cos(x ) so that dv = sin(x )dx and du = dx . Then, x sin(x )dx = = u dv uv โ v du
cos(x )dx = โx cos(x ) + = โx cos(x ) + sin(x ) + C Thomas Scanlon Math 16b Example Integrate: xe x dx
Take u = x and v = e x so that du = dx and dv = e x dx . x e x dx = = = = u dv uv โ xe x โ v du e x dx xe x โ e x + C Thomas Scanlon Math 16b Another example Integrate e x cos(x )dx
Set u = e x and v = sin(x ) so that du = e x dx and dv = cos(x )dx . Then e x cos(x )dx = = = = u dv uv โ v du e x sin(x )dx e x sin(x )dx e x sin(x ) โ e x sin(x ) โ Thomas Scanlon Math 16b Solution, continued e x cos(x )dx = e x sin(x ) โ e x sin(x )dx e x cos(x )dx e x cos(x )dx e x cos(x )dx = e x sin(x ) + e x cos(x ) โ e x cos(x )dx = e x sin(x ) + e x cos(x ) โ
2 e x cos(x )dx = e x sin(x ) + e x cos(x ) Thomas Scanlon Math 16b 1 1 2 2 x and y = cos(x ) so that dw = e x dx and Set w = e dy = โ sin(x )dx Then
x e x cos(x )dx = e x sin(x ) + e x cos(x ) + C A third example Integrate ln(x )dx Set u = ln(x ) and v = x so that du = ln(x )dx =
= = = =
Thomas Scanlon Math 16b dx x and dv = dx . u dv uv โ v du x dx x dx
1 x ln(x ) โ x ln(x ) โ x ln(x ) โ x + C ...
View
Full Document
 Fall '06
 Sarason
 Calculus, Chain Rule, Integration By Substitution, The Chain Rule, dx, Thomas Scanlon, Thomas Scanlon Math

Click to edit the document details